Find the values of a and b in a limit

[songoku]

Homework Statement

Please see below

Relevant Equations

Limit

L'Hopital Rule is not allowed

I know $$\lim_{h\rightarrow 0} af(h)+bf(2h)−f(0)=0$$
$$a+b=1$$

But I don't know how to find the second equation involving a and b. I imagine I need to somehow obtain $h$ in numerator so I can cross out with $h$ in denominator but I don't have idea how to get $h$ in the numerator.

Thanks

 

[Office_Shredder]

Try using Taylor's theorem to express $f(h)$ as a linear polynomial in $h$ plus an error term that decreases superlinearly, and do the same for $f(2h)$.

 

[FactChecker]

Try using Taylor's theorem to express $f(h)$ as a linear polynomial in $h$ plus an error term that decreases superlinearly, and do the same for $f(2h)$.

Based on the definition of derivative that he gave in another thread, I think this is not the approach that is expected.

 

[songoku]

Based on the definition of derivative that he gave in another thread, I think this is not the approach that is expected.

Yes, my lesson has not covered Taylor's theorem

Try using Taylor's theorem to express $f(h)$ as a linear polynomial in $h$ plus an error term that decreases superlinearly, and do the same for $f(2h)$.

I tried using Taylor's theorem, as suggested, as best as I can and by only taking the term up until $h$ I get $a+2b=0$

Is there other possible method to solve the question without using Taylor?

Thanks

 

[PeroK]

Yes, my lesson has not covered Taylor's theorem


I tried using Taylor's theorem, as suggested, as best as I can and by only taking the term up until $h$ I get $a+2b=0$

Is there other possible method to solve the question without using Taylor?

Thanks

Why not do the obvious decomposition of the limit into derivatives of $f$?

 

[songoku]

Why not do the obvious decomposition of the limit into derivatives of $f$?

I think I understand this hint.

Thank you very much for the help Office_Shredder, FactChecker, PeroK

 

[SammyS]

Why not do the obvious decomposition of the limit into derivatives of $f$?

I think I understand this hint.

Just in case you don't quite understand that hint, or the advice from @FactChecker in Post #3 ...

You had the following as one of the definitions for $f'(x_0)$.

$\displaystyle f'(x_0)=\lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$

So that $\displaystyle \ f'(0)=\lim_{h \to 0} \frac{f(h)-f(0)}{h} \ .$

You may not realize that $\displaystyle \ \lim_{h \to 0} \frac{f(2h)-f(0)}{2h} = f'(0)\$ as well.

(Added a short time later with the Edit feature):

In decomposing the given limit, take the $\displaystyle \frac{b\,f(2h)}{h}$ term, for instance and write it as $\displaystyle \frac{b\,(f(2h)-f(0))}{h} + \frac{b\,(f(0))}{h} \ .$

Do similarly with $\displaystyle \frac{a\,f(h)}{h}$, combine like terms, etc.